No, your homeowner's policy doesn't have anything to do with the damage you caused with your car. A homeowner's insurance policy actually excludes automobiles being covered on the policy. It wouldn't cover the liability if you damaged someone else's property or hurt them with your car. It also wouldn't cover your car if your garage caught on fire or somebody stole the vehicle. You need auto insurance to cover your car and its operation.
The damage you caused will be covered under the liability portion of your auto insurance policy. There is no deductible you need to pay when it comes to liability. If you damaged your car by hitting their light than that would go under your collision coverage, if you carry that on your auto insurance policy, with whatever deductible you selected.
Depending on your insurance company your rate may increase with an accident such as this. Some insurance companies won't increase your rate if the dollar amount paid out is under some threshold like $600. You might also have accident forgiveness on your policy where your rates won't go up with one accident. If the damage is minor and your neighbor agrees to it you can also just pay out of pocket for the damage to the light and don't even make a claim on your insurance policy.